Summary: H. Abdo et al. [MATCH Commun. Math. Comput. Chem. 72, No. 3, 741–751 (2014; Zbl 1464.05049)] demonstrated that there exist connected graphs for which \(\mu^2(G)\approx M_2(G)\) where \(\mu(G)\) is the spectral radius of a graph \(G\), \(M_2(G)\) is the second Zagreb index and \(m\) the number of edges. We use and extend this approximation to investigate opportunities to convert results from spectral graph theory into results involving the first and the second Zagreb indices \(M_1(G)\) and \(M_2(G)\). We do this principally by noting that \(M_1(G)=w_3\) and \(2M_2(G)=w_4\), where \(w_3\) and \(w_4\) are the numbers of 3- and 4-walks in a graph, respectively.